↑

Properties

Label 2-1050-35.33-c1-0-4
Degree $2$
Conductor $1050$
Sign $0.922 - 0.385i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 + 0.499i)4-s + i·6-s + (−2.19 − 1.48i)7-s + (0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (0.366 + 0.633i)11-s + (0.965 − 0.258i)12-s + (−1.03 + 1.03i)13-s + (−0.866 + 2.49i)14-s + (0.500 − 0.866i)16-s + (0.586 − 2.19i)17-s + (0.258 − 0.965i)18-s + (−2.09 + 3.63i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.557 − 0.149i)3-s + (−0.433 + 0.249i)4-s + 0.408i·6-s + (−0.827 − 0.560i)7-s + (0.249 + 0.249i)8-s + (0.288 + 0.166i)9-s + (0.110 + 0.191i)11-s + (0.278 − 0.0747i)12-s + (−0.287 + 0.287i)13-s + (−0.231 + 0.668i)14-s + (0.125 − 0.216i)16-s + (0.142 − 0.531i)17-s + (0.0610 − 0.227i)18-s + (−0.481 + 0.833i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.922 - 0.385i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.922 - 0.385i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7046732562\)
\(L(\frac12)\) \(\approx\) \(0.7046732562\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (2.19 + 1.48i)T \)
good11 \( 1 + (-0.366 - 0.633i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.03 - 1.03i)T - 13iT^{2} \)
17 \( 1 + (-0.586 + 2.19i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.09 - 3.63i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.60 - 0.965i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 8.19iT - 29T^{2} \)
31 \( 1 + (-6.86 + 3.96i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.22 - 4.57i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.46iT - 41T^{2} \)
43 \( 1 + (-0.138 - 0.138i)T + 43iT^{2} \)
47 \( 1 + (-10.6 + 2.84i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.36 - 5.08i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.267 + 0.464i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.5 - 6.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.86 - 1.03i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + (-7.72 - 2.07i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.03 - 4.06i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.24 - 8.24i)T - 83iT^{2} \)
89 \( 1 + (3.23 - 5.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.94 - 4.94i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.05914168680676458296085413130, −9.461375956116504913282769692872, −8.392454955834519203409993890371, −7.41141746119428215252383560264, −6.64066256834892525016571656314, −5.69811834817641480561769112185, −4.54136169729784295974407156404, −3.71835566254617654308353305387, −2.51627578151184366924147848763, −1.09198388319321500295533909063, 0.43166284668963841133879670993, 2.42442961499939216834761930595, 3.81018234974733768556240831367, 4.80705619358338821472637685045, 5.87531040883317572084558368977, 6.29364777844404091293604478760, 7.22236666102856302451955179354, 8.239749689859440192289603615448, 8.997459518572973561566220433524, 9.863844237488603702699265062727

Graph of the $Z$-function along the critical line